A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g., comprising part of, one, or several dies) on a substrate (e.g., a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at one time, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
In order to monitor the lithographic process, parameters of the patterned substrate are measured. Parameters may include, for example, the overlay error between successive layers formed in or on the patterned substrate and critical linewidth of developed photosensitive resist. This measurement may be performed on a product substrate and/or on a dedicated metrology target. There are various techniques for making measurements of the microscopic structures formed in lithographic processes, including the use of scanning electron microscopes and various specialized tools. A fast and non-invasive form of specialized inspection tool is a scatterometer in which a beam of radiation is directed onto a target on the surface of the substrate and properties of the scattered or reflected beam are measured. By comparing the properties of the beam before and after it has been reflected or scattered by the substrate, the properties of the substrate can be determined. This can be done, for example, by comparing the reflected beam with data stored in a library of known measurements associated with known substrate properties. Two main types of scatterometer are known. Spectroscopic scatterometers direct a broadband radiation beam onto the substrate and measure the spectrum (intensity as a function of wavelength) of the radiation scattered into a particular narrow angular range. Angularly resolved scatterometers use a monochromatic radiation beam and measure the intensity of the scattered radiation as a function of angle.
Typically, properties of the substrate such as overlay errors on a wafer are measured across the exposure field and across the wafer substrate. Overlay errors are not controlled at each and every measurement point. Rather, a parameterization is used to characterize overlay across the field and wafer.
This parameterization describes the overlay errors as a function of wafer and field position. This parameterization can be simple, such as a 10-parameter model, or more complex, such as a basis set of high order polynomials. Using this parameterization, the corrections can be calculated that should be applied at the lithography apparatus with the aim of driving overlay errors across the wafer (field) to zero.
The precision of the calculated control move is a function of the variance of the measurements and the model. The model is determined by where measurements are performed, so the precision of the control move is ultimately a function of the variance of the measurements, the model, and where measurements are performed.
If the basis function that defines the model is non-orthogonal across the wafer (field) domain, there will then be linear dependence between the parameters, which makes the system non-unique and the parameters harder to estimate. As a result, control precision is adversely impacted.